On the adaptive deterministic block Kaczmarz method with momentum for solving large-scale consistent linear systems

Abstract

The Kaczmarz method is a traditional and widely used iterative algorithm for solving large-scale consistent linear systems, while its improved block Kaczmarz-type methods have received much attention and research in recent years due to their excellent numerical performance. Hence, in this paper, we present a deterministic block Kaczmarz method with momentum, which is based on Polyak’s heavy ball method and a row selection criterion for a set of block-controlled indices defined by the Euclidean norm of the residual vector. The proposed method does not need to compute the pseudo-inverses of a row submatrix at each iteration and it adaptively selects and updates the set of block control indices, thus this is different from the block Kaczmarz-type methods that are based on projection and pre-partitioning of row indices. The theoretical analysis of the proposed method shows that it converges linearly to the unique least-norm solutions of the consistent linear systems. Numerical experiments demonstrate that the deterministic block Kaczmarz method with momentum method is more efficient than the existing block Kaczmarz-type methods.